Defences and You

Posted: July 24, 2011 by qqwerthy in Guides, Items, Metagame

Last week I did an article for entry level players. This week we are going to meta-game, hard.

Armor and magic resistance are stats that decrease the damage you receive. That part is easy.

But you need to maximize your damage. That’s the hard part, and a question a few people in our gaming group have asked is, when do items like Last Whisper and Void Staff pay off?

Since I’m a mathematical guy I figured I’d go over this.

To start I’d like to go over the formula for Armor and Magic resist, so you can better understand how they work.
The formula is the same for both Armor and MR.

Damage Reduction = 100/(100 + x)

With x being the value of the Armor or MR.

The graph looks like this:

With the x axis being the value of your armor or resistance, and the y axis is the percent(%) damage taken.
As you can see there is a diminishing return on defense, meaning it takes more and more to obtain lower and lower percentages of damage taken.

Important points to note are:
0 defense = 100% damage taken or no damage reduction
33 defense = 75% damage taken or 25% damage reduction
100 defense = 50% damage taken or 50% damage reduction
300 defense = 25% damage taken or 75% damage reduction

*Most tanks don’t get to 300 defense, please keep this in mind. The graph is extended only to show the point of diminishing returns.

So as you can see going from 0 to 150 armor gives you more damage reduction than going from 150 to 300.

Now here is where things get interesting. Defense can go into the negatives, with armor/mr  reducing/penetrating items.

“If a target has negative armor, the amount of bonus damage is equal to the amount of damage reduction caused by that positive amount of armor. This sounds confusing and is better illustrated with an example.
•    If a target has 10 armor, they take 9.1% reduced damage (formula from Armor).
◦    Therefore a target with -10 armor would take 9.1% bonus damage.
◦    So if you attack for 100 physical damage on the target, it would take 109.1 damage instead.”

⁃    -(direct quote from http://leagueoflegends.wikia.com)

“When attacking, armor penetration and armor reduction are applied to the target champion’s armor in the following order:
1.    Percentage Armor Reduction
2.    Flat Armor Reduction
3.    Flat Armor Penetration
4.    Percentage Armor Penetration”    -(again sourced from lol wikia)

The same facts hold true for MR.

So now that we know all this lets start looking at some penetration items to maximize our damage.

We’ll start with magic penetration items first.

1.  Percent MR reduction items: spoilers there are none. The only way to get % MR reduction is through champion abilities. So this category is out.

2. Flat MR reduction items:
 Abyssal Scepter: UNIQUE Aura: Reduces the Magic Resist of nearby enemy champions by 20, +70 AP, +57 Magic Resistance. 2650 Gold.

Malady: UNIQUE: Your physical attacks shred your target doing 20 magic damage and reducing their magic resistance by 6. Stacks up to 4 times. +25 AP, 50% Attack Speed. 1825 Gold.
For our purposes we are going to assume that you are an AP carry and thus malady is not an optimal choice.

3. Flat MR penetration items:
Haunting Guise: UNIQUE Passive: +20 Magic Penetration, +25 AP, +200 Health. 1485 Gold.

Sorcerer’s Shoes: +20 Magic Penetration 1100 Gold.

4. Percentage MR Reduction:
Void Staff: UNIQUE Passive: Spells penetrate 40% of the defender’s magic resist, +70 AP. 2295 Gold.

It is also worth noting in this section that there is the 9 point talent in offense mastery tree Archaic Knowledge: Your
magical spells penetrate 15% of the target’s magic resistance.

Now there are also runes to calculate in. Assuming you are a serious AP carry, you probably have a full set of Red Insight Magic Pen runes. This nets you 8.5 Magic Pen. I know there are also Quints for this, but quints tend to be a personal preference type of thing.

So assuming the 8.5 pen from runes, the 15% from masteries, and assuming you take Sorcerer’s Shoes over Merc treads, let’s compare your damage graph to the original.

Your new equation looks something like:

Opponent’s Damage reduction = 100/ (100 +  .85(x-28.5))

As you can see you are clearly doing much more damage than before and you are doing more damage for longer
But the topic of this article is at what resistance should you consider getting an item like Void Staff.

By adding the 40% magic penetration to your item build, your opponents equation now looks like this

Opponent’s Damage reduction = 100 / (100 + .51(x-28.5))
Now let’s add that to the graph

As you can see its value is not that great at lower resistances, but at around the 100 MR mark is when it really starts to pay off.

So now lets extrapolate this:

“At level 18, Magic Resistance ranges from 30 (most champions) to 54 ( Malphite and  Yorick) (23 percent and 35 percent damage reduction, respectively).” -source lol wikia

So if a champion builds zero MR, then the max you will find is 54, at which point Void staff is worthless.
But MR is cheap, and a lot of players build Merc treads and eventually banshee’s veil. So we are easily talking about +75 bonus MR, putting you well over 100 MR, at which point Void staff increases your damage by 15% on top of having a decent AP bonus. And if the tank on the enemy team is stacking more MR you will keep doing solid damage.

It is my opinion that because of these facts, Void staff is unparalleled in its effective increase of Damage output. It should be a core item of every AP carry, built as soon as they see 2 or more of the enemy team building anything with a Negatron Cloak.

Now let’s look at Abyssal Scepter and Haunting Guise for a moment. Both of these offer 20 penetration. I say penetration because when you simplify the equation penetration and reduction are treated as equal. So back to the point, both offer 20 penetration. And penetration is good right, it increases my damage right? Yes but these items are exceptionally situational.

Abyssal Scepter is an interesting item because it serves 3 functions.

  • It has a decent amount of AP which increases damage.
  • It provides MR which is good defensively to keep you alive.
  • It also has 20 magic reduction which will also increase your damage.

I like building Abyssal Scepters either very early, or very late in the game. I say this because if you build it early then you give yourself an extra MR advantage over your opponents and you have more Magic pen which might push them into the negatives (bonus damage). Or I build it as a last spot item when my build is full and I have extra money and a little more AP and a little more resist would be nice.

Haunting Guise is not as interesting though, it clearly has no idea what it wants to do as an item.

  • It gives 25 AP which increases damage.
  • 200 health which can be nice but health isn’t that good of a defense unless you stack insane amounts of it.
  • And it gives 20 magic penetration.

Personally I think building a haunting guise is a waste of time and money and here’s why. For 1485 gold it is mediocre at everything it tries to do, and builds into nothing better. For 115 gold more than the total cost of a haunting guise you can get a Needlessly Large Rod and start building a Death cap or a Zhonya’s Hourglass. Both of which are better for damage I feel, and Zhonya’s is vastly superior defensively than 200 hp.

TL;DR Void staff is amazing, abyssal scepter is okay, haunting guise sucks.

Now lets look at Armor Penetration items.

1. Percent Armor Reduction items: Again there are none. % armor reduction comes solely from player abilities.

2. Flat Armor Reduction items:
Stark’s Fervor: +20% Attack Speed UNIQUE Aura: Gives nearby allied Champions 20% Lifesteal, 20% Attack Speed, and 30 Hp5. Reduces the Armor of nearby enemy champions by 20. 2550 Gold.
The Black Cleaver: +55 Attack Damage +30% Attack Speed UNIQUE Passive: Physical attacks reduce your target’s Armor by 15 for 5 seconds, (maximum 3 stacks). 2865 Gold.

3. Flat Armor Penetration items:
The Brutalizer: +25 Attack Damage UNIQUE Passive: Reduces ability cooldowns by 10%, +15 Armor Penetration. 1337 Gold.
Youmuu’s Ghostblade: +30 Attack Damage +15% Critical Chance UNIQUE Passive: +20 Armor Penetration, 15% Cooldown Reduction. Unique Active: You gain +20% movement speed and +50% attack speed for 4 seconds. Attacking enemy units with melee attacks increases the duration by 2 to a maximum of 8 seconds. 60 second cooldown. 2687 Gold.
Sword of the Divine: +60% Attack Speed UNIQUE Passive: Every fourth attack deals 100 extra Magic Damage. UNIQUE Active: Your attacks cannot be dodged and you gain 30 Armor Penetration for 8 sec. 1970 Gold.

4. Percent Armor Penetration:
Last Whisper: +40 Attack Damage UNIQUE: +40% Armor Penetration 2290 Gold.

Now we must also note Runes and Masteries. You can get up to 6 armor penetration from talents.
Also a full set of red Desolation runes will give you an additional 14.94 armor pen. And this time I am going to include Desolation quints because they are so damn good for an additional 9.99 armor pen. Total of 30.93 armor penetration at level 1.

So assuming the 30.93 penetration from runes and masteries and no other sources or abilities, let’s compare your damage graph to the original.

Your new equation looks something like:

Damage reduction = 100 / (100 +(x-30.93)

That is a lot of bonus damage and it is all achievable at level 1. Essenstially your opponent must get an additional 31 armor just to break even at lower levels and subsequently armor values. If they did this they would be significantly lowering their own damage.

So now let’s look at some items.

First up is Sword of the Divine. Unless you are Twisted Fate or the enemy team has a professional Jax player, you probably aren’t building Sword of the Divine. And even if you are for whatever reason, I hope it isn’t for the Armor Pen because an 8 second buff is not worth 1970 gold just to get Armor Pen. So lets look at better items.

Second item to consider is The Brutalizer. A buddy of mine swears by this item. Personally, unless you plan to build Youmuu’s, I don’t think it is worth it to build it for damage or armor pen, maybe its a nice niche item if you need a little damage boost and some cooldown reduction, but other than that its a no go. For the 1337 gold it costs you could just save up and buy a BF Sword to increase your damage and then build into…

The Black Cleaver. Now this is an amazing item. At 3 stacks, the targets armor is reduced by 45. And that’s not the best part, it reduces the armor for everyone, not just for your damage. So if you are playing a Ranged AD carry and you get this bad boy, all your bruisers that haven’t gotten it yet will benefit.
So add 45 to the already 30.93 penetration you have and you get something that looks like this

Opponent’s Damage reduction = 100/ (100 + (x-75.93)

With the black cleaver you will be doing insane amounts of bonus damage. Your opponent must get 76 armor just to take 100% damage and no bonus damage. That means they need even more armor to start reducing your damage. It is a ridiculously powerful item.

Now look at this
“At level 18, armor ranges from a base of  Heimerdinger’s 61 armor (38% reduced damage), to  Poppy’s and  Shen’s 90 armor (47% reduced damage).” -(source lol wikia)

So if a character at level 18 builds zero armor, with runes masteries and black cleaver, you will most likely be doing full or bonus damage. If they build armor they better build a lot because you will still be tearing them to shreds.

Stark’s Fervor is a hard item to place. It tries to do a lot and is pretty expensive for what it does. The aura bonus is only good if there are 3 AD champions on your team. Now since this is No Elo and we don’t subscribe to high end meta or professional play, you might very well find yourself on a team with 2 AD ranged carries and a bruiser. If this is the case, I’d say one person should get a Stark’s Fervor. It is a decent item for what it does, but I would not get it if I was the only one going to benefit from it. Just make another Bloodthirster if that is the case.

Last, but not least, the item of much controversy in my group of gaming friends, Last Whisper. You would think that because it is so similar to the Void Staff in price and design that it would be an amazing item. You would be wrong.

The following graph depicts the difference in damage of a character who went Last Whisper first and a character who went Black cleaver.

Even though Last Whisper is cheaper, it does not outperform the Black Cleaver until the target has over 150 armor. Let us recall that unless your enemy specifically builds armor they won’t have more than 90. So even if the squishiest of AP casters or support built a Zhonya’s Hourglass (armor 50), the Most armor they would have would be 111, they would need to build an additional tank item before Last Whisper would be useful. And that also means everyone on the team needs 2 tank items for it to be effective.

It is my opinion that unless the whole enemy team consists of 5 tanks all building straight armor, Last Whisper is wholly useless. It gives less damage, no attack speed, and less penetration than Black Cleaver.

It is also my opinion that unless you are playing an AD carry with abilities that synergize well with critically strike ( ex. Ashe, Tryndamere, Gangplank, etc…) that you should rush Black Cleaver before Infinite Edge. Because of negative armor it should increase your damage much more.

TL;DR Stark’s is situational, Black Cleaver is amazing and should always be rushed, Last Whisperer should never be built.

Special thanks this article goes out to the fine people at http://leagueoflegends.wikia.com
from where I got all my graphics and information.

Disagree with me? Have a question? Think my math is wrong? Please leave a comment below I’d be happy to discuss anything. And if my math is wrong, please for the love of god correct me, the last thing I want is for someone to take wrong math seriously.

PS. To all the AD carries out there, try rushing Black Cleaver instead of Infinity edge and tell me about the differences.

Comments
  1. John Petrina says:

    My real question is about AD champs who run bloodrazor/tank i.e. WW. Or champs like teemo and kog maw who run bloodrazor and have an on hit magic damage effect. Is it worth it to build a malady for the mag pen, to make the bloodrazor really sing? Should I be specing mag pen? Or should that be considered extra and I want to go AR pen like normal?

    So I guess my question is Mag Pen VS Ar Pen, in respect to bloodrazor/wits end/malady. (I know it’s pretty champ specific.)

  2. qqwerthy says:

    Great question, and also a very difficult one to answer.

    There is no clear cut way to measure those AttackSpeed + on hit effect AD champs compared to pure AD damage champs because they are dealing 2 separate sources of damage, both physical and magic which are subject to separate resistances.

    I don’t have an solid math for you at the moment to tell you what is clear cut and best. But personally whenever I play Teemo or Kog’maw, I typically either rush Witt’s End if I can buy it in one shot on my first go back to spawn, or I buy parts of Malady since it is a decent item.

    I’d recommend still going Armor Pen, because you can get more of it from just straight runes and masteries, which means you will naturally be doing more and it will still work nicely with all that attackspeed.

    Also check out this video, it is slightly related to the question you asked and a good watch.

  3. Phil says:

    If you look at armor and magic resist in terms of effective hp, there is no diminish in returns. Regardless of how much armor or magic resist you already have, each point in the stat will increase your effective hp by 1% against the corresponding type of damage.
    http://leagueoflegends.wikia.com/wiki/Armor#Diminishing_Returns

    ^That explains it a little better than I can.

    Also, the main difference between reduction and penetration is that reduction can go negative while penetration can’t.

  4. Tiago Dias says:

    hey there. i find this very interesting but i have a question.

    since this order applies:

    1. Percentage Armor Reduction
    2. Flat Armor Reduction
    3. Flat Armor Penetration
    4. Percentage Armor Penetration

    if my target has 40 armor and i have 30 armor pen + black cleaver, you are assuming that his armor will get to -35. but if we count black cleaver as flat armor reduction and armor pen as flat armor penetration. and the armor penetration CAN’T reduce armor below 0, shouldn’t it be:

    1. 40 armor
    2. -5 armor
    3. -5 armor ( no effect by armor penetration)

    instead of:

    1. 40 armor
    2. -5 armor
    3. -35 armor

    correct me if i’m wrong xD

    btw first time in this website. congratz for the article.

  5. qqwerthy says:

    You are absolutely correct Tiago, the first example you gave would be correct.

    Do to limitations in the graphing program I used, I was unable to graph a function with different cases. I decided to leave the graphs as is, instead of cutting them off, because for most part you would rarely be hitting negatives, and because the real focus of the article is on the break points of when the different items are better than one another.

  6. […] Staff – Qqwerthy’s article covers just how awesome this item is. For characters with less than a 1 to 1 ratio on AP, this item […]

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